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16-735 Paper Presentation “Numerical Potential Field Techniques for Robot Path Planning” †

16-735 Paper Presentation “Numerical Potential Field Techniques for Robot Path Planning” †. Sept, 19, 2007 NSH 3211 Hyun Soo Park, Iacopo Gentilini.

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16-735 Paper Presentation “Numerical Potential Field Techniques for Robot Path Planning” †

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  1. 16-735 Paper Presentation“Numerical Potential Field Techniques for Robot Path Planning” † Sept, 19, 2007 NSH 3211 Hyun Soo Park, IacopoGentilini † Barraquand, J., Langlois, B., and Latombe, J.-C.IEEE Transactions on Systems, Man and CyberneticsVolume 22, Issue 2, Mar/Apr 1992 , pages: 224 - 241 Robotic Motion Planning 16-735 Potential Field Techniques

  2. How to generate collision free paths? 1. Global approach: • building a “connectivity graph” of collision free configuration • searching the graph for a path (e.g. network of one dimensional curves) Image from “Numerical Potential Field Techniques for Robot Path Planning” 2. Local approach: - searching a grid placed across the robot’s configuration using heuristic functions (e.g. tangent bug, potential field) Robotic Motion Planning 16-735 Potential Field Techniques

  3. Differences between global and local? 1. Global approach: • advantages: very quick search in the “connectivity graph” • disadvantages: expensive precomputation step to get the graph (exponential in the dimension n of the configuration space Qwhere n is number of the robot’s degrees of freedom) 2. Local approach: • advantages: no precomputation needed • disadvantages: - “search graph” considerably larger than “connectivity graph” - dead ends (local minima) Robotic Motion Planning 16-735 Potential Field Techniques

  4. How to combine advantages of both? • Incrementally build a graph connecting the local minima of potential functions defined over the configuration space  (no expensive precomputation) • Concurrently searching this graph until the goal is reached → escaping local minima (search within much smaller “search graph”) • Based on multiscale pyramids of bitmap arrays of and (not analytically defined potential function) Robotic Motion Planning 16-735 Potential Field Techniques

  5. Basic functions • k-neighborhood with k  [1,r ] of a point x in a grid of dimension r is defined as the set of points in the grid having at most k coordinates differing from those of x: • k = 1  2 r points • k = 2  2 r2 points • k = r  3r -1points 1. Forward kinematic:X : R × Q → W (p, q) ↦ x = X (p, q) where p R is a point in the robot • Workspace bitmap:BM :W→(1,0) • x↦BM (x) where BM(x) = 0 represents free • is discretized in a 2D or 3D - dimensional grid  and  free: • - a 1-neighborhood is used, that means 4 neighbors in 2D, 6 neighbors in 3D, and 2n neighbors in n-D within the discrete grid; • - preparation: a “wavefront” expansion is computed by setting each point in freeneighbor of boundary or of ito 1; than the neighbors of this new points to 2 and so on until all free has been explored; 1- neighborhood (2D) Robotic Motion Planning 16-735 Potential Field Techniques 5

  6. Multiscale pyramid - workspace representation is given as a grid at a 512×512 level of resolution; using a scaling factor 2 a pyramid of representations is also computed until the coarsest resolution level 16×16 is reached: -  is the distance between two adjacent points ( min = 1/512 and  max = 1/16 if given in percentage of the workspace diameter) g g  s s g g  s Robotic Motion Planning 16-735 Potential Field Techniques

  7. Basic functions • Configurationspace:Qisalso discretized in a n-dimensional grid GQand GQfree • - the resolution is defined as the logarithm of the inverse of the distance between two discretizaton points - the resolution r of GW is also: • - for any given workspace resolution r, the corrisponding resolution Ri of the discretization of Q along the qi axis is chosen in such a way that a modification of qi by Δigenerates a small motion of the robot (any pointp of R moves less than nbtol×  ) : • where: Robotic Motion Planning 16-735 Potential Field Techniques 7

  8. How are potential functions built? W-potential: - computed in W Q-potential: - defined over Q where G is called the arbitration function - good Q-potential in (whose dimension is big) - if Vpiare free of local minimawe can not assume that U is free of local minima: it depends on thedefinition of G where pi are the controlpoints in the robot R - small dimension of (2 or 3) for low cost information - have to be built such that they are free of local minima (neededprecomputation) Image from “Numerical Potential Field Techniques for Robot Path Planning” Robotic Motion Planning 16-735 Potential Field Techniques

  9. W-Potential 1. Simple W-Potential: • get the position of control point p in and its goal position xgoal • set Vp = 0 at xgoal • set the neighbors in  free of xgoalto 1 and so on Image from “Numerical Potential Field Techniques for Robot Path Planning” -Vpis the direction to goal 2. Improved W-Potential: • build the workspace skeleton S as subset of  freecomputing the “wavefront” expansion • connect xgoal to and compute Vpin the augmented S using a queue of points of S sorted by decreasing value • compute Vp in  free\ as shownin 1. Image from “Numerical Potential Field Techniques for Robot Path Planning” Image from “Numerical Potential Field Techniques for Robot Path Planning” Robotic Motion Planning 16-735 Potential Field Techniques

  10. Q-Potential • attracts control points pitoward their respective goal position • arbitration function definition (minimize local minima!): • - concurrent attraction causes local minima - avoid zero value when one point have reached the goal Robotic Motion Planning 16-735 Potential Field Techniques

  11. Techniques to construct local-minima graph • Best First motion • Random motion • Valley-guided motion • Constrained motion Robotic Motion Planning 16-735 Potential Field Techniques

  12. Valley Guided Motion Technique Images from “Google Map 2007” Robotic Motion Planning 16-735 Potential Field Techniques

  13. Valley Guided Motion Technique Searching valleys V of Q-potential U in Qfree • using -U calculated in qstart and qgoal reach to local minima qi and qg • search V for a path connecting qsand qg. Atevery crossroad a decision is made using anheuristic function defined as Q-potential Uheur • if step b. is successful, path is calculated,otherwise failure qstart qs qg V Best experimental Q-potential function: qgoal where s is a small number Robotic Motion Planning 16-735 Potential Field Techniques 13

  14. Valley Guided Motion Technique When a point q Q is a valley points (qV)? • compute U(q); • compute the 2n values of U at the 1-neihbors of q; • for each possible valley direction i [1,n] • compare U(q) to the 2n – 2 values of U at the 1-neighbors in the hyperplane orthogonal to the qi axis • if U(q) is smaller or equal to these 2n – 2 values, q is a valley point. q n = 2 - complexity is O(n2) or if using 2-neighborhood O(n4)- better using n-neighborhood but cardinals are 3n-1 with exponential complexity Robotic Motion Planning 16-735 Potential Field Techniques 14

  15. Constrained Motion Technique Starting from qstart in Qfree • follow -U flow until local minima qloc is attained; • if qloc = qgoal the problem is solved; otherwise execute a “constrained” motion +Mi(qloc) or -Mi(qloc) with i [1,n] : • increase iteratively the ith configuration space coordinate by the increment Δiuntil a saddle point of the local minimum well is reached (U decreases again). If (q1,…, qi, …,qn) is the current configuration its successor minimizes U over the set consisting of (q1,…, qi+ Δi ,…,qn) and its 1-neighbors in thehyperplane orthogonal to the qi axis (the motion thus track a valley in the (n-1)-dimensional subspace orthogonal to the qi axis). • terminate the constrained motion and execute an other gradient motion; qloc n = 2 Q-potential function used: Robotic Motion Planning 16-735 Potential Field Techniques 15

  16. Best First Motion and Random Motion Technique 1. Best-First Motion Technique 2. Random Motion Technique Agitation Robotic Motion Planning 16-735 Potential Field Techniques

  17. Best First Motion andRandom Motion Technique 1. Best-First Motion Technique • - Good for n <= 4 • What if n is getting bigger? •  Searching unit increases in almost exponential order ( ) as increasing DOF •  Thus, we need another algorithm to search local minima 2. Random Motion Technique - The number of iteration can be specified by user so that this algorithm performs fast. Robotic Motion Planning 16-735 Potential Field Techniques

  18. Random Motion Technique Local Minimum Detection Limited number of searching iteration If U(q) > U(q’), then q’ is successor  Gradient motion If NO q’, then q is local minimum Robotic Motion Planning 16-735 Potential Field Techniques

  19. Random Motion Technique Path Joining Adjacent Local Minima Smoothing This can be performed concurrently on a parallel computer because of no need to communicate between the different processing unit  Random motion Robotic Motion Planning 16-735 Potential Field Techniques

  20. Random Motion Technique Dead-end No more local minima near current position Backtrack to arbitrary point in line of to which is selected by uniform distribution law. Then try to find another local minima. Drawback : No guarantee to find a path whenever one exists. However, by property of Brownian Motion, as the number of iteration of random motion, Robotic Motion Planning 16-735 Potential Field Techniques

  21. Random Motion Technique PDF for Brownian Motion can be described as Gaussian Distribution Function Probability of location of qiafter time t (end of random motion) Robotic Motion Planning 16-735 Potential Field Techniques

  22. Random Motion Technique Duration of Random Motion Should not be too long  waste of time and no gradient motion Should not be too short  No chance to escape Robotic Motion Planning 16-735 Potential Field Techniques

  23. Random Motion Technique Duration of Random Motion Attraction Radius ( ) No correlation between potential function and attraction radius  Only probabilistic result can be obtained. Mean Distance Robotic Motion Planning 16-735 Potential Field Techniques

  24. Random Motion Technique Duration of Random Motion Since attraction radius can’t exceed workspace diameter, by normalizing it to 1, we can obtain, By taking the supremum of the Jacobian matrix, we can guarantee more coarse motion in workspace. Robotic Motion Planning 16-735 Potential Field Techniques

  25. Random Motion Technique Duration of Random Motion Since attraction radius is strongly positive, we treat it as random variable with truncated Laplace Distribution of density. Finally, we have Double exponential shape of Laplace Distribution of density Due to At first we actually assume attraction radius is constant which is the same as diameter of workspace. However, if we have distribution function of attraction radius we also can have corresponding probability of t. Expectation Robotic Motion Planning 16-735 Potential Field Techniques

  26. Conclusion Approach : - Constructing a potential field over the robot’s configuration - Building a graph connecting the local minima of the potential - Searching graph Aim : Escaping local minima 4 techniques : - Best-first motion : gives excellent result with few DOF robots (n < 5) - Random motion : gives good results with many DOF - Valley-Guided motion : inferior result but can be improved in future - Constrainted motion : good at planning the coordinated motions Robotic Motion Planning 16-735 Potential Field Techniques

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